Calculation of Biological Assays

Bioassays are methods employed to estimate the effect of a given substance in living matter, and therefore they are frequently used in the pharmaceutical industry. This is essential for the determination of potency and the assurance of activity of many proteins, vaccines, complex mixtures, and products for cell and gene therapy, as well as for their role in monitoring the stability of biological products.

Application of Bioassays

  • Process Development
  • Process Characterization
  • Product Release
  • Process Intermediates
  • Stability
  • Qualification of Reagents
  • Product Integrity


A set of steps that will help guide the analysis of a bioassay:

1. As a part of the chosen analysis, select the subset of data to be used in the determination of the relative potency using the pre-specified scheme. Exclude only data known to result from technical problems such as contaminated wells, non-monotonic concentration–response curves, etc.

2. Fit the statistical model for detection of potential outliers, as chosen during development, including any weighting and transformation. This is done first without assuming similarity of the Test and Standard curves but should include important elements of the design structure, ideally using a model that makes fewer assumptions about the functional form of the response than the model used to assess similarity.

3. Determine which potential outliers are to be removed and fit the model to be used for suitability assessment. Usually, an investigation of outlier cause takes place before outlier removal. Some assay systems can make use of a statistical (non-investigative) outlier removal rule, but removal on this basis should be rare. One approach to “rare” is to choose the outlier rule so that the expected number of false positive outlier identifications is no more than one; e.g., use a 1% test if the sample size is about 100. If a large number of outliers are found above that expected from the rule used, that calls into question the assay.

4. Assess system suitability. System suitability assesses whether the assay Standard preparation and any controls behaved in a manner consistent with past performance of the assay. If an assay (or a run) fails system suitability, the entire assay (or run) is discarded and no results are reported other than that the assay (or run) failed. Assessment of system suitability usually includes adequacy of the fit of the model used to assess similarity. For linear models, adequacy of the model may include assessment of the linearity of the Standard curve. If the suitability criterion for linearity of the Standard is not met, the exclusion of one or more extreme concentrations may result in the criterion being met. Examples of other possible system suitability criteria include background, positive controls, max/min, max/background, slope, IC50 (or EC50), and variation around the fitted model.

5. Assess sample suitability for each Test sample. This is done to confirm that the data for each Test sample satisfy necessary assumptions. If a Test sample fails sample suitability, results for that sample are reported as “Fails Sample Suitability.” Relative potencies for other Test samples in the assay may still be reported. Most prominent of sample suitability criteria is similarity, whether parallelism for parallel models or equivalence of intercepts for slope-ratio models. For nonlinear models, similarity assessment involves all curve parameters other than EC50 (or IC50).

6. For those Test samples in the assay that meet the criterion for similarity to the Standard (i.e., sufficiently similar concentration–response curves or similar straight-line subsets of concentrations), calculate relative potency estimates assuming similarity between Test and Standard, i.e., by analyzing the Test and Standard data together using a model constrained to have exactly parallel lines or curves, or equal intercepts.

7. A single assay is often not sufficient to achieve a reportable value, and potency results from multiple assays can be combined into a single potency estimate. Repeat steps 1–6 multiple times, as specified in the assay protocol or monograph, before determining a final estimate of potency and a confidence interval.

8. Construct a variance estimate and a measure of uncertainty of the potency estimate (e.g., confidence interval).


Bioassays Analysis Models

1. Quantitative and Qualitative Assay Responses

2. Parallel-Line Models for Quantitative Responses

3. Nonlinear Models for Quantitative Responses

4. Slope-Ratio Concentration–Response Models

5. Dichotomous (Quantal) Assays


Relative Potency Calculation

A primary assumption underlying methods used for the calculation of relative potency is that of similarity. Two preparations are similar if they contain the same effective constituent or same effective constituents in the same proportions. If this condition holds, the Test preparation behaves as a dilution (or concentration) of the Standard preparation. Similarity can be represented mathematically as follows.

Let FT be the concentration–response function for the Test, and let FS be the concentration–response function for the Standard. The underlying mathematical model for similarity is:

FT(z) = FS(ρ z)

Where, z represents the concentration and ρ represents the relative potency of the Test sample relative to the Standard sample.


Parallel-Line Concentration Response Models

If the general concentration–response model (Quantitative and Qualitative Assay Responses) can be made linear in x = log (z), the resulting equation is then:

y = α + βlog(z) + e = α + βx + e,

Where, e is the residual or error term, and the intercept, α, and slope, β, will differ between Test and Standard.


With the parallelism (equal slopes) assumption, the model becomes –

yS = α + βlog(z) + e = αS + βx + e               [3.2]

yT = α + βlog(ρz) + e = [α + βlog(ρ)] + βx + e = αT + βx + e,

Where S denotes Standard, T denotes Test, αS = α is the y-intercept for the Standard, and αT = α + βlog (ρ) is the y-intercept for the Test.


Nonlinear Models for Quantitative Responses

Nonlinear concentration–response models are typically S-shaped functions. They occur when the range of concentrations is wide enough so that responses are constrained by upper and lower asymptotes. The most common of these models is the four-parameter logistic function as given below.

Let y denote the observed response and z the concentration. One form of the four-parameter logistic model is -

One alternative, but equivalent, form is –

The two forms correspond as follows:

Lower asymptote: D = a0

Upper asymptote: A = a0 + d

Steepness: B = M (related to the slope of the curve at the EC50)

Effective concentration 50% (EC50): C = antilog (b) (may also be termed ED50).


Parallel-Curve Concentration–Response Models

Log ρ is the log of the relative potency and the horizontal distance between the two curves, just as for the parallel-line model.


Slope-Ratio Concentration–Response Models

If a straight-line regression fits the non-transformed concentration–response data well, a slope-ratio model may be used. The equations for the slope-ratio model assuming similarity are then:

yS = α + βz + e = α + βSz + e

yT = α + β(ρz) + e = α + βSρz + e = α + βTz + e


The model consists of one common intercept, a slope for the Test sample results, and a slope for the Standard sample results as in equation. The relative potency is then found from the ratio of the slopes:

Relative Potency = Test sample slope/Standard sample slope = βρ/β = ρ


Dichotomous (Quantal) Assays

The logit model for the probability of response, P(z), can be expressed in two equivalent forms. For the sigmoid,

Where log (ED50) = − β0/β1. An alternative form shows the relationship to linear models:

Utilizing the parameters estimated by software, which include β0, β1, and β2 and their standard errors, one obtains the estimate of the natural log of the relative potency:

Combining Independent Assays (Sample-Based Confidence Interval Methods)

Let Ri denote the logarithm of the relative potency of the ith assay of N assay results to be combined. To combine the N results, the mean, standard deviation, and standard error of the Ri are calculated in the usual way:

A variety of statistical methods can be used to analyze bioassay data. This article presents several methods, but many other similar methods could also be employed. Additional information and alternative procedures can be found in the listed below and other sources:

1. Bliss CI. The Statistics of Bioassay. New York: Academic Press; 1952.

2. Bliss CI. Analysis of the biological assays in U.S.P. XV. Drug Stand. 1956;24:33–67.

3. Böhrer A. One-sided and two-sided critical values for Dixon’s outlier test for sample sizes up to n = 30. Econ Quality Control. 2008;23:5–13.

4. Brown F, Mire-Sluis A, eds. The Design and Analysis of Potency Assays for Biotechnology Products. New York: Karger; 2002.

5. Callahan JD, Sajjadi NC. Testing the null hypothesis for a specified difference—the right way to test for parallelism. Bioprocessing J. 2003:2;71–78.

6. DeLean A, Munson PJ, Rodbard D. Simultaneous analysis of families of sigmoidal curves: application to bioassay, radioligand assay, and physiological dose–response curves. Am J Physiol. 1978;235:E97–E102.

7. European Directorate for the Quality of Medicines. European Pharmacopoeia, Chapter 5.3, Statistical Analysis. Strasburg, France: EDQM; 2004:473–507.

8. Finney DJ. Probit Analysis. 3rd ed. Cambridge: Cambridge University Press; 1971.

9. Finney DJ. Statistical Method in Biological Assay. 3rd ed. London: Griffin; 1978.

10. Govindarajulu Z. Statistical Techniques in Bioassay. 2nd ed. New York: Karger; 2001.

11. Hauck WW, Capen RC, Callahan JD, et al. Assessing parallelism prior to determining relative potency. PDA J Pharm Sci Technol. 2005;59:127–137.

12. Hewitt W. Microbiological Assay for Pharmaceutical Analysis: A Rational Approach. New York: Interpharm/CRC; 2004.

13. Higgins KM, Davidian M, Chew G, Burge H. The effect of serial dilution error on calibration inference in immunoassay. Biometrics. 1998;54:19–32.

14. Hurlbert, SH. Pseudo replication and the design of ecological field experiments. Ecological Monogr. 1984;54:187–211.

15. Iglewicz B, Hoaglin DC. How to Detect and Handle Outliers. Milwaukee, WI: Quality Press; 1993.

16. Nelder JA, Wedderburn RWM. Generalized linear models. J Royal Statistical Soc, Series A. 1972;135:370–384.

17. Rorabacher DB. Statistical treatment for rejection of deviant values: critical values of Dixon’s “Q” parameter and related subrange ratios at the 95% confidence level. Anal Chem. 1991;63:39–48.

Related: Biological Assay Validation


Reference: USP-NF 1032, 1034

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